CN110244767B - Formation configuration reconstruction optimization method adopting finite element method - Google Patents
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Abstract
The invention discloses formation configuration reconstruction optimization by adopting a finite element method, which divides formation reconstruction time into a series of equal-interval intervals by utilizing the finite element method, converts a reconstruction path solving problem into an optimization problem, further overcomes the limitation problem of the traditional reconstruction method, and realizes the reconstruction of satellite formation. In the optimization process, according to a numerical iteration method, the optimal orbit control quantity and the corresponding reconstructed orbit state quantity in each time subinterval are determined, and the control acceleration loading and the reconstructed orbit state quantity applied to the spacecraft by the small-thrust engine in the optional subinterval are iteratively corrected again by considering the minimum criterion of the total fuel consumption control, the control opportunity, the safety factor for avoiding collision between the spacecrafts and the like.
Description
Technical Field
The invention relates to the technical field of satellite formation flight, in particular to formation configuration reconstruction optimization by adopting a finite element method.
Background
As a multi-satellite system, the formation of the spacecraft greatly expands the limitation of a single spacecraft in completing tasks, not only can the task function of the single spacecraft in the formation be realized, but also the whole formation can replace the single large spacecraft to realize more complex tasks. Under certain conditions, even if one or two spacecrafts are lost in the formation, other spacecrafts can cooperate to complete the task of losing the spacecrafts, so that the overall reliability of the formation system is greatly improved, and the formation system has a very high practical application value. Because the fuel carried by the satellites in the formation is limited and can not be supplied, the research on different reconstruction strategies for formation flight has very important significance.
In order to successfully achieve formation flight mission, spacecraft formation orbit design before and during mission is particularly important, thus posing problems associated with formation configuration. The formation organization mainly comprises three main contents:
building a configuration, namely building mutually independent spacecraft groups into a configuration required by a task;
configuration keeping, namely, each spacecraft in the formation can still keep a stable relative position under the influence of factors such as orbit perturbation and the like through a control method, and the configuration required by a task is maintained;
and (2) configuration reconfiguration, namely, for the reconfiguration of the positions of the flying spacecrafts in formation, the orbit of one or more spacecrafts in the formation of the spacecrafts is adjusted according to the requirements of different tasks, so that the formation positions of the spacecrafts or the relative positions of the spacecrafts in the formation are changed, and the conversion between different configurations of the formation is realized.
Finite element method (abbreviated as finite element method) is a structural analysis numerical method that divides a structural grid into computational models. Through popularization and development, the method becomes an approximate method for solving a mathematical physical equation. The finite element method is actually based on the mathematical variation principle, the node configuration is relatively arbitrary, the calculation format is more complex than that of the finite difference method, and the required computer memory amount is also relatively large. The finite element method is commonly used in aircraft structural analysis, and is a conventional analysis method, no matter static strength analysis, dynamic strength analysis, fatigue and fracture or thermal strength analysis. In recent years, as research becomes deeper and deeper, the application of the finite element method in other fields is also more and more extensive.
Compared with the traditional spacecraft orbit maintenance, the orbit control of high-precision formation flight has higher requirements on control precision and certain limits on control frequency. Therefore, the continuous low-thrust orbit control enters the visual field of the astronaut, and the maintenance and reconstruction of the spacecraft orbit based on the continuous low-thrust control become popular research contents of a plurality of learners. As a common key technology in the deep space exploration task, the low-thrust technology has the characteristics of mature technical development, high control precision and intuition. The application of the continuous low-thrust method is wider due to the wider and wider application range of the novel electric propeller and the electromagnetic force and the electrostatic force.
In addition, in the reconstruction of the satellite formation, the total fuel consumption, the fuel consumption balance or some weighted relation of the two is generally used as the optimization index of the formation reconstruction.
Disclosure of Invention
In view of the limitation of the traditional method on the reconstruction of the satellite formation, the invention adopts a finite element method to divide the reconstruction time into a series of equal interval intervals, and applies orbit control to the reconstruction time element nodes on the basis of a CW motion equation of the relative motion of the satellite so as to avoid the collision in the series of time intervals as a constraint condition, thereby converting the reconstruction problem into an optimization problem. According to a numerical iteration method, the optimal track control quantity and the corresponding reconstructed track in each time subinterval are determined, and iterative correction is performed on the track control quantity and the reconstructed track according to the fuel optimal criterion, the control opportunity and the collision condition, so that the reconstruction configuration is completed quickly, the overall reliability of the formation system is improved, and the fuel consumption is reduced.
The invention relates to a formation configuration reconstruction optimization method adopting a finite element method, which comprises the following steps:
establishing a relative motion model of a main satellite and a slave satellite under a circle shooting-free orbit;
the formation of satellites consists of a master satellite and a slave satellite, and the master satellite is assumed to run on a circle-free orbit. In order to maintain the formation flight configuration, a CW equation is adopted to describe the relative position relationship of the master satellite and the slave satellite, and external control acceleration is added in the CW equation.
Relative motion relationship of the master star and the slave star:
dividing the reconstruction time into subintervals, and solving the reconstruction equation obtained in the first step by adopting a finite element method to obtain a control matrix;
in the invention, the node value x of the state variable x (τ) of the spacecraft varies with timek，xk+1Used to determine the reconstruction trajectory and the control quantities, are considered as optimized quantities in the reconstruction problem. The invention integrates elements of the reconstruction time interval by a finite element method to obtain a system set with 6M dimensions of a linear equation set:
step three, taking the minimum total fuel consumption as a convergence constraint, ensuring the avoidance of collision in the whole reconstruction process, and optimizing and solving the control of formation reconstruction;
in the invention, find the control U provided by the small thrust engine applied on the slave star in the reconstruction process of the formation configurationi(i-1, 2, …, N is the total number of spacecraft in the formation) is the main goal of reconstruction optimization, achieving minimum fuel consumption under the constraint of collision avoidance. The sum of the squared magnitudes of the control acceleration vectors is selected as the performance indicator, which essentially corresponds to the lowest fuel delivery quantity C.
The safety of the spacecraft is ensured in the formation reconstruction process by applying the method, so that the distance for keeping the safety of the spacecraft is taken as a constraint condition.
The invention adopts the formation configuration reconstruction optimization of the finite element method and has the advantages that:
the invention introduces a finite element method in structural mechanics into a relative orbit configuration reconstruction research, can realize satellite formation reconstruction by optimizing and solving orbit control on finite nodes of formation reconstruction time intervals, and greatly reduces reconstruction path optimization time.
Secondly, the minimization method adopted by the invention always starts with the optimal track of each spacecraft, solves the initial optimal solution and finds the final optimal solution, and avoids collision of the formation satellites.
And thirdly, the total fuel consumption is used as an optimization variable, the selection of independent variables in the reconstruction problem is converted into a nonlinear optimization problem with equality constraint, the energy consumption can be greatly reduced by the solved reconstruction path, and the overall service life of the satellite formation is prolonged.
Drawings
FIG. 1 is a schematic diagram of the relative positions of two stars in formation flight.
Fig. 2 is a schematic diagram of flight formation collision avoidance.
Fig. 3 is a diagram of a trajectory before and after reconstruction and a reconstruction path in embodiment 1 of the present invention.
Fig. 4 is a time history of small thrust acceleration during reconstruction in embodiment 1 of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
The basic idea of the finite element method is to discretize a continuous solution area into a finite set of element combinations that are connected to each other in a certain way. In the present invention, the detailed related contents of the finite element method refer to the contents of the brief course of finite element, the author ZhaoQu, etc., pages 1-3, on the 1 st edition of 9 months 2009.
The invention relates to a formation configuration reconstruction optimization method adopting a finite element method, which comprises the following steps:
establishing a relative motion model of a main satellite and a slave satellite under a circle shooting-free orbit;
the formation of satellites consists of a master satellite and a slave satellite, and the master satellite is assumed to run on a circle-free orbit. The primary star orbital coordinate system shown with reference to fig. 1 is defined as follows: the principal star is marked as S1From star as S2The origin is the center of mass of the main satellite, the x axis deviates from the geocentric, the z axis is vertical to the orbit plane of the satellite, and the y axis is in the orbit plane of the satellite. The distance vector from the geocentric O to the centroid of the main star is recorded as r1The distance vector from the centroid O to the centroid of the star is denoted as r2And the distance between the centroid of the master star and the centroid of the slave star is recorded as rho. In order to maintain the formation flight configuration, a CW equation is adopted to describe the relative position relationship between the master satellite and the slave satellite, and an external control acceleration u is added in the CW equation, so that the master satellite and the slave satellite satisfy the relative motion relationship of the formula (1):
x represents the Slave star S2Relative to the main star S1The motion state quantity of (2).
Phi represents a coefficient matrix belonging to X, namely a relative motion model of the main satellite and the auxiliary satellite under the circle-shooting-free orbit, and is referred to as a motion coefficient matrix for short.
u denotes the control acceleration applied to the slave star (acceleration loading for short) provided by the low thrust engine.
A represents the lower right diagonal element of phi, and
n represents the orbital angular velocity of the main satellite; -A represents the inverse of A.
B represents the lower left diagonal element of phi,
b represents the inverse of B.
In the present invention, the CW equation refers to "spacecraft formation flight" 1 st edition of month 1 in 2015: dynamics, controls, and navigation, author (U.S.) Kyle T. Alfriend et al, Zhang Shijie translation, pages 57-58.
Jordan decomposition V on Φ-1J, V denotes the eigenvector of the Jordan decomposition, V-1Expressing the inverse matrix of V to obtain a characteristic matrix J characterizing six-dimensional motion mode, wherein
Let Z be V-1X, Z denote the motion state quantities after coordinate transformation (simply referred to as transformed motion state quantities) belonging to X, X denotes the satellite S2Relative to the main star S1The motion state quantity of (2).
Considering the problem of formation reconstruction consisting of N spacecraft within a fixed time interval [0, T ] of the reconstruction time T, the constraints being to avoid collision of the spacecraft with each other and to achieve minimum fuel consumption, the initial and final positions and velocities of the spacecraft have been given. The relative motion relationship can be obtained by bringing J into formula (1):
i denotes the number of the spacecraft in the formation, i ═ 1,2, …, N;
Zirepresenting the motion state quantity after the coordinate transformation of the ith spacecraft;
uirepresenting a control acceleration applied to the ith spacecraft;
p represents the coefficient of u, and the value of P is the elements of column 3 to column 6 portions of V.
The derivation of equation (2) and the decoupling of the 6 motion components yields a state variable that is summarized in the form of reconstructed equation (3):
i denotes the number of the spacecraft in the formation, i ═ 1,2, …, N;
j represents the state variable serial number of the single spacecraft, j is 1,2, …, 6;
xi,ja jth state variable representing an ith spacecraft;
λ represents the coefficient of the first order of the reconstruction equation, and λ -n when the dominant star is on the circular relative orbit2。
ui,jRepresenting the control acceleration applied to the jth state variable of the ith spacecraft.
In the present invention, equation (3) is applied to summarize the reconstruction of the state variables to solve the formation track reconstruction path solution problem using a finite element method.
Dividing the reconstruction time into subintervals, and solving the reconstruction equation obtained in the first step by adopting a finite element method to obtain a control matrix;
the present invention uses a finite element method to solve the reconstruction equation (3). Reconstruction time interval [0, T ] from initial orbit configuration to target orbit configuration of star]The decomposition is M elements, namely the subintervals of the variable field, and the first time node is marked as t1，t 10, the second time node is marked t2The kth time node is marked as tkK ∈ M, k denotes the reconstruction time interval [0, T]The identification number of the time node in is located at the tkThe previous time node is denoted as tk-1At said tkThe following time node is denoted as tk+1Since the division into M intervals results in tM+1T. These elements may be of different lengths of time depending on the requirements of the reconstruction problem or the numerical accuracy requirements achieved for reconstructing the tracks. The elements of the reconstruction interval are connected with each other through nodes, namely the starting point and the end point of each subintervalThe end is a node, and the node also provides a place for the low thrust engine to control the track.
Considering the weighted residual, defining a finite element weight coefficient ω (τ), ignoring the subscript, one can obtain:
k denotes the sequence number of M elements of the reconstruction time interval, k is 0,1, …, M;
τ represents an arbitrary time in the kth element of the reconstruction interval;
tka kth node representing a reconstruction time interval;
tk+1a (k + 1) th node representing a reconstruction time interval;
ω (τ) represents a finite element weight coefficient over time;
λ (τ) represents a coefficient of a first order term that varies with time, and λ (τ) — n when the main star is located on a circular opposite orbit2。
x (τ) represents a state variable of the spacecraft over time;
u (τ) represents the control acceleration load exerted by the small thrust engine on the spacecraft as a function of time;
d τ is expressed as the differential in time.
In particular, two interpolation functions in the kth element of the reconstruction interval
And
defined by lagrange polynomials:
the value of the state variable x (τ) of the spacecraft, which varies with time in the kth element, can be represented by x(k)(τ) to approximate:
x(k)(τ) represents the mean state variable of the spacecraft in the kth element;
xkstate variables representing the spacecraft on the kth node of the reconstruction time interval;
xk+1state variables of the spacecraft on the k +1 th node of the reconstruction time interval are represented.
State variable x(k)(τ) varies linearly with time τ. Thus, the approximate reconstructed orbit in each cell will be a line segment and the entire reconstructed orbit from the star will be a segmented line. As the time grid becomes denser, the segmented line will approach the true trajectory of the formation reconstruction problem.
λ (τ) achieves the approximation using the same processing method as x (τ):
λ(k)(τ) represents coefficients of first order terms in the kth element;
λka first order coefficient at a kth node representing a reconstruction time interval;
λk+1the first order term coefficients at the k +1 th node representing the reconstruction time interval.
Reconstruction time interval [0, T]The control maneuver provided by the internal element internal low-thrust engine is distributed at the node time tkAnd tk+1Of the two velocity pulses in (a), which can be represented by a dirac function,
u(k)(τ) represents the control acceleration loading applied to the spacecraft in the kth element;
in order to apply the finite element method to the reconstruction time interval [0, T ] in the whole reconstruction orbit]In the following, the focus will be on obtaining a set of equations involving all the node values in the grid and their corresponding control maneuvers. In a finite element tk,tk+1]And (3) integrating the formula (4), and constructing a finite element equation aiming at the state variable x as follows:
wherein the content of the first and second substances,
p represents the row sequence number of the element position of the control matrix, and 1 or 2 is taken;
q represents the column sequence number of the element position of the control matrix, and 1 or 2 is taken;
since control is performed at each node, the other nodes, except the first and last, are comprised of control maneuvers provided by two parts of a low thrust engine. Therefore, the total acceleration load Δ u on the kth nodekCan be obtained by the formula (10)
ΔukRepresenting the total acceleration load provided by the low thrust engine at the kth node;
however, equation (10) does not hold for the first and last nodes. The reason is that the first and last nodes are located at the boundary of the entire formation reconstruction process, and only one calculation is performed. For this purpose, the initial and final velocities located in the first and last elements need to be corrected: in the first element, the initial velocity v of the formation slave stars0Is modified into
In the last element, the final velocity v of the formation slave starsfMust be changed into
Wherein the content of the first and second substances,
representing the initial moment of the 0 th node to control acceleration loading;
indicating the end time of the (M-1) th element to control acceleration loading.
In the invention, the node value x of the state variable x (τ) of the spacecraft varies with timek，xk+1Used to determine the reconstruction trajectory and the control quantities, are considered as optimized quantities in the reconstruction problem. The invention integrates elements of a reconstruction time interval by a finite element method to obtain a system set with 6M dimensions of a linear equation set in the form of formula (11):
the control acceleration loading performed on the first and last nodes can be obtained according to equation (12):
step three, taking the minimum total fuel consumption as a convergence constraint, ensuring the avoidance of collision in the whole reconstruction process, and optimizing and solving the control of formation reconstruction;
finding the control U provided by the low-thrust engine applied to the slave stars during the reconstruction of the formation configurationi(i-1, 2, …, N is the total number of spacecraft in the formation) is the main goal of reconstruction optimization, achieving minimum fuel consumption under the constraint of collision avoidance. The sum of the squared magnitudes of the control acceleration vectors is selected as a performance indicator, which essentially corresponds to the lowest fuel delivery quantity C:
wherein, Ui,kIs the control provided by the low thrust engine at the kth time node of the ith spacecraft, is Deltaui,kThe inverse transformed form of (a); u shapei,k TThe upper corner mark T in (a) is a coordinate transposer.
N is the total number of the spacecrafts in the formation;
m is the number of elements for decomposing the reconstruction time interval from the initial orbit configuration to the target orbit configuration of the satellite;
Δui,kis the total acceleration load provided by the low thrust engine on the kth time node of the ith spacecraft.
In addition, in the process of formation and reconstruction, the safety of the spacecraft should be ensured, so the distance for keeping the safety of the spacecraft is taken as a constraint condition. In order to avoid collisions between the spacecraft, an exclusion sphere centered on the spacecraft is defined and it is mandatory that these spheres do not intersect, and that at most one point of coincidence between two spheres is accepted in the reconstruction time. As shown in FIG. 2, the safe distance is denoted as R, assuming that the safe distance around each spacecraft is half the radius
During the reconstruction process, in addition to one tangent point, a plurality of spacecraft (such as spacecraft 1, spacecraft 2, spacecraft 3 in fig. 2) should ensure thatAre not intersected.
Thus, the reconstruction problem is transformed into an optimization problem by the finite element method. The method of the invention starts with the optimal trajectory of each spacecraft, without considering the collision risk. By ignoring the constraints of avoiding collisions, an initial optimal solution for the lowest fuel delivery quantity C is found. After that, the calculated optimal solution is used as the initial value of the track reconstruction optimization problem to find the final solution of the lowest fuel transfer quantity C and consider avoiding collision.
Example 1
As shown in fig. 3, the formation of satellites consists of one master star and five slave stars (i.e., slave star 1, slave star 2, slave star 3, slave star 4, slave star 5). The reference orbit of the dominant star is given by the orbit elements: the semi-major axis of the track is 29948.478 km, the eccentricity is 0.001, the inclination angle of the track is 63.31 degrees, the ascension at the ascending crossing point is 243.00 degrees, the breadth angle at the perigee is 214 degrees, and the mean perigee angle is 180 degrees. Suppose that the five slave stars in the formation are initially distributed along the orbital direction, centred on the master star, 50m apart. The five slave star initial states are as follows:
The position and velocity components are in units of m and m/s, and the five satellite final positions and velocities are designed as follows:
The simulation results are shown in fig. 3, and fig. 4 shows the fuel consumption of each slave star, taking into account the number of 15 elements. The simulation result verifies the effectiveness of the finite element method in the reconstruction problem, and the finite element method does not need too much calculation time, so that the method can realize the reconstruction path planning from the initial track configuration to the target track configuration, and the position coordinates of the reconstruction path do not have large-scale deviation with the initial track target track. This indicates that the slave star and the master star do not generate a large relative distance in the reconstruction process, and can still be maintained within the formation range, and the CW equation is still established.
Claims (2)
1. A formation configuration reconstruction optimization adopting a finite element method is characterized in that the formation configuration reconstruction comprises the following steps:
establishing a relative motion model of a main satellite and a slave satellite under a circle shooting-free orbit;
the formation of the satellites consists of a master satellite and a slave satellite, and the master satellite is supposed to run on a non-shooting circle orbit, so that the master satellite and the slave satellite meet the relative motion relation of the formula (1):
x represents the Slave star S2Relative to the main star S1The motion state quantity of (2);
phi represents a motion coefficient matrix;
u represents the acceleration loading;
a represents the lower right diagonal element of phi, and
n represents the orbital angular velocity of the main satellite; -a represents the inverse of a;
b represents the lower left diagonal element of phi,
-B represents the inverse of B;
jordan decomposition V on Φ-1J, V denotes the eigenvector of the Jordan decomposition, V-1Expressing the inverse matrix of V to obtain a characteristic matrix J characterizing six-dimensional motion mode, wherein
Let Z be V-1X, Z represents the motion state quantity after transformation, X represents the satellite S2Relative to the main star S1The motion state quantity of (2);
considering the formation reconstruction problem consisting of N spacecrafts within a fixed time interval [0, T ] of reconstruction time T, wherein the constraint conditions are to avoid collision among the spacecrafts and realize minimum fuel consumption, and the initial and final positions and speeds of the spacecrafts are given; the relative motion relationship can be obtained by bringing J into formula (1):
i denotes the number of the spacecraft in the formation, i ═ 1,2, …, N;
Zirepresenting the motion state quantity after the coordinate transformation of the ith spacecraft;
uirepresenting a control acceleration applied to the ith spacecraft;
p represents a coefficient of u, and the value of P is the elements of the 3 rd column to the 6 th column of V;
the derivation of equation (2) and the decoupling of the 6 motion components yields a state variable that is summarized in the form of reconstructed equation (3):
i denotes the number of the spacecraft in the formation, i ═ 1,2, …, N;
j represents the state variable serial number of the single spacecraft, j is 1,2, …, 6;
xi,ja jth state variable representing an ith spacecraft;
lambda denotes reconstructionCoefficient of the first order of the equation, λ ═ n when the dominant star is on the circular relative orbit2；
ui,jRepresenting the control acceleration applied to the jth state variable of the ith spacecraft;
dividing the reconstruction time into subintervals, and solving the reconstruction equation obtained in the first step by adopting a finite element method to obtain a control matrix;
considering the weighted residual, a finite element weight coefficient ω (τ) is defined:
k denotes the sequence number of M elements of the reconstruction time interval, k is 0,1, …, M;
τ represents an arbitrary time in the kth element of the reconstruction interval;
tka kth node representing a reconstruction time interval;
tk+1a (k + 1) th node representing a reconstruction time interval;
ω (τ) represents a finite element weight coefficient over time;
λ (τ) represents a coefficient of a first order term that varies with time, and λ (τ) — n when the main star is located on a circular opposite orbit2；
x (τ) represents a state variable of the spacecraft over time;
u (τ) represents the control acceleration load exerted by the small thrust engine on the spacecraft as a function of time;
d τ is expressed as the differential of time;
two interpolation functions in the kth element of the reconstruction interval
And
defined by lagrange polynomials:
the value of the state variable x (τ) of the spacecraft, which varies with time in the kth element, can be represented by x(k)(τ) to approximate:
x(k)(τ) represents the mean state variable of the spacecraft in the kth element;
xkstate variables representing the spacecraft on the kth node of the reconstruction time interval;
xk+1representing the state variable of the spacecraft on the (k + 1) th node of the reconstruction time interval;
state variable x(k)(τ) varies linearly with time τ; thus, the approximate reconstructed orbit in each cell will be a line segment, and the entire reconstructed orbit from the star will be a segment line; as the time grid becomes denser, the segmented line will approach the true trajectory of the formation reconstruction problem;
λ (τ) achieves the approximation using the same processing method as x (τ):
λ(k)(τ) represents coefficients of first order terms in the kth element;
λka first order coefficient at a kth node representing a reconstruction time interval;
λk+1a first order coefficient at the (k + 1) th node representing a reconstruction time interval;
reconstruction time interval [0, T]The control maneuver provided by the internal element internal low-thrust engine is distributed at the node time tkAnd tk+1Of the two velocity pulses in (a), which can be represented by a dirac function,
u(k)(τ) represents the control acceleration loading applied to the spacecraft in the kth element;
in order to apply the finite element method to the reconstruction time interval [0, T ] in the whole reconstruction orbit]In the following, we will focus on obtaining a set of equations relating all the node values in the grid and their corresponding control maneuvers; in a finite element tk,tk+1]And (3) integrating the formula (4), and constructing a finite element equation aiming at the state variable x as follows:
wherein the content of the first and second substances,
p represents the row sequence number of the element position of the control matrix, and 1 or 2 is taken;
q represents the column sequence number of the element position of the control matrix, and 1 or 2 is taken;
since control is performed at each node, except for the first node and the last node, the other nodes are all composed of control maneuvers provided by two parts of low-thrust engines; therefore, the total acceleration load Δ u on the kth nodekCan be obtained by the formula (10)
ΔukOn the kth nodeThe total acceleration load provided by the low thrust engine;
however, equation (10) does not hold for the first and last nodes; the reason is that the first and last nodes are located at the boundary of the whole formation reconstruction process and only once calculation is carried out; for this purpose, the initial and final velocities located in the first and last elements need to be corrected: in the first element, the initial velocity v of the formation slave stars0Is modified into
In the last element, the final velocity v of the formation slave starsfMust be changed into
Wherein the content of the first and second substances,
representing the initial moment of the 0 th node to control acceleration loading;
representing the ending time of the M-1 th element to control acceleration loading;
node value x of a time-varying state variable x (τ) of a spacecraftk，xk+1The method is used for determining a reconstruction orbit and a control quantity, and the reconstruction orbit and the control quantity are regarded as an optimized quantity in a reconstruction problem; integrating the elements of the reconstruction time interval by a finite element method to obtain a system set with 6M dimensions of a linear equation set in the form of a formula (11):
the control acceleration loading performed on the first and last nodes can be obtained according to equation (12):
step three, taking the minimum total fuel consumption as a convergence constraint, ensuring the avoidance of collision in the whole reconstruction process, and optimizing and solving the control of formation reconstruction;
finding the control U provided by the low-thrust engine applied to the slave stars during the reconstruction of the formation configurationiIs the main goal of reconstruction optimization, and realizes the minimization of fuel consumption under the constraint of avoiding collision; the sum of the squares of the magnitudes of the control acceleration vectors is selected as the performance index, and the sum of the squares of the magnitudes of the control acceleration vectors is calculated at a value corresponding to the lowest fuel delivery amount C:
wherein, Ui,kIs the control provided by the low thrust engine at the kth time node of the ith spacecraft, is Deltaui,kThe inverse transformed form of (a); u shapei,k TThe upper corner mark T in the middle is a coordinate transpose symbol;
n is the total number of the spacecrafts in the formation;
m is the number of elements for decomposing the reconstruction time interval from the initial orbit configuration to the target orbit configuration of the satellite;
Δui,kis the total acceleration load provided by the low thrust engine on the kth time node of the ith spacecraft.
2. The formation configuration reconfiguration optimization using finite element method according to claim 1, wherein: reconstruction time interval [0, T ] from initial orbit configuration to target orbit configuration of star]Decomposed into M elements, i.e. variable fieldsSubinterval, the first time node marked t1，t10, the second time node is marked t2The kth time node is marked as tkK ∈ M, k denotes the reconstruction time interval [0, T]The identification number of the time node in is located at the tkThe previous time node is denoted as tk-1At said tkThe following time node is denoted as tk+1Since the division into M intervals results in tM+1＝T。
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